On the non-existence of perfect codes in the Niederreiter-Rosenbloom-Tsfasman metric

In this paper we consider codes in \(\mathbb{F}_q^{s\times r}\) with packing radius \(R\) regarding the NRT-metric (i.e. when the underlying poset is a disjoint union of chains with the same length) and we establish necessary condition on the parameters \(s,r\) and \(R\) for the existence of perfect...

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Bibliographic Details
Published in:arXiv.org 2023-05
Main Authors: Gubitosi, Viviana, Portela, Aldo, Qureshi, Claudio
Format: Article
Language:English
Subjects:
Knapsack problem
Set theory
Online Access:Get full text
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Summary:In this paper we consider codes in \(\mathbb{F}_q^{s\times r}\) with packing radius \(R\) regarding the NRT-metric (i.e. when the underlying poset is a disjoint union of chains with the same length) and we establish necessary condition on the parameters \(s,r\) and \(R\) for the existence of perfect codes. More explicitly, for \(r,s\geq 2\) and \(R\geq 1\) we prove that if there is a non-trivial perfect code then \((r+1)(R+1)\leq rs\). We also explore a connection to the knapsack problem and establish a correspondence between perfect codes with \(r>R\) and those with \(r=R\). Using this correspondence we prove the non-existence of non-trivial perfect codes also for \(s=R+2\).
ISSN:2331-8422